Mathematical Proof and the Principles of Mathematics/Logic/Rules of inference summary

This is a list of the rules of inference given in previous sections. In the notation being used, a solid horizontal bar means that the statement below is a valid deduction from the statement(s) above. A vertical bar with a horizontal bar connected to it means that whatever is to the right of the vertical bar is a subproof, and whatever is above the horizontal bar are assumption(s) and whetever is below the horizontal bar is what has been derived. The names given are just placeholders and no guarantees are made that they are standard in any way.

Propositional logic

Rules not requiring subproofs

• Iteration

$P$

$P$

• Use of contradiction

$False$

$P$

• Disjunction by first case

$P$

$P$ or $Q$

• Disjunction by second case

$Q$

$P$ or $Q$

• First use of conjunction

$P$ and $Q$

$P$

• Second use of conjunction

$P$ and $Q$

$Q$

• Implication from the conclusion

$Q$

$P$ implies $Q$

• Implication from false assumption

not $P$

$P$ implies $Q$

• Double negation

not not $P$

$P$

• Equivalence to implication

$P$ iff $Q$

$P$ implies $Q$

• Equivalence to converse

$P$ iff $Q$

$Q$ implies $P$

• Conjunction by components

$P$

$Q$

$P$ and $Q$

• Use of disjunction, first alternative false

$P$ or $Q$

not $P$

$Q$

• Use of disjunction, second alternative false

$P$ or $Q$

not $Q$

$P$

• Use of implication, from premise (modus ponens)

$P$ implies $Q$

$P$

$Q$

• Use of implication, from false conclusion (modus tollens)

$P$ implies $Q$

not $Q$

not $P$

Rules requiring one subproof

• Implication by direct proof

$P$ $Q$

$P$ implies $Q$